Approximation and composition of functions in quantized tensor trains via orthogonal polynomial expansions

Abstract:This work explores the representation of univariate and multivariate functions as quantized tensor trains (QTT). It develops a constructive algorithm that employs expansions in orthogonal polynomial bases and Clenshaw evaluations to represent analytic and highly differentiable functions as QTT approximants. This approach provides a general framework for function composition in QTT form that generalizes efficiently to multidimensional settings. Without loss of generality, the presentation focuses on the Chebyshev basis, where the method demonstrates rapid convergence for highly differentiable functions, in agreement with classical theoretical predictions, and remains numerically stable at high polynomial orders. The performance of the algorithm is compared with that of tensor cross-interpolation (TCI) and multiscale interpolative constructions, demonstrating competitive performance and favorable scaling behavior in certain scenarios.

19 pages, 16 figures. Published in Linear Algebra and its Applications

Quantum Physics (quant-ph); Numerical Analysis (math.NA)

arXiv:2407.09609 [quant-ph]

(or arXiv:2407.09609v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2407.09609 Focus to learn more arXiv-issued DOI via DataCite

https://doi.org/10.1016/j.laa.2026.08.022 Focus to learn more DOI(s) linking to related resources

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.